Optimal. Leaf size=222 \[ \frac{59 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{2 x^2+2}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{1050 \sqrt{x^4+3 x^2+2}}-\frac{3 \sqrt{x^4+3 x^2+2} x}{175 \left (5 x^2+7\right )}+\frac{1}{75} \sqrt{x^4+3 x^2+2} x+\frac{9 \left (x^2+2\right ) x}{175 \sqrt{x^4+3 x^2+2}}-\frac{9 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{175 \sqrt{x^4+3 x^2+2}}+\frac{9 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{2 x^2+2}} \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{2450 \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.444338, antiderivative size = 333, normalized size of antiderivative = 1.5, number of steps used = 21, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {1228, 1099, 1135, 1122, 1189, 1223, 1716, 1214, 1456, 539} \[ -\frac{3 \sqrt{x^4+3 x^2+2} x}{175 \left (5 x^2+7\right )}+\frac{1}{75} \sqrt{x^4+3 x^2+2} x+\frac{9 \left (x^2+2\right ) x}{175 \sqrt{x^4+3 x^2+2}}+\frac{44 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{1875 \sqrt{x^4+3 x^2+2}}+\frac{81 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{8750 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{9 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{175 \sqrt{x^4+3 x^2+2}}+\frac{3 \sqrt{2} \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}}-\frac{39 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{12250 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1099
Rule 1135
Rule 1122
Rule 1189
Rule 1223
Rule 1716
Rule 1214
Rule 1456
Rule 539
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2+x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx &=\int \left (\frac{52}{625 \sqrt{2+3 x^2+x^4}}+\frac{16 x^2}{125 \sqrt{2+3 x^2+x^4}}+\frac{x^4}{25 \sqrt{2+3 x^2+x^4}}+\frac{36}{625 \left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}}-\frac{12}{625 \left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}}\right ) \, dx\\ &=-\left (\frac{12}{625} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx\right )+\frac{1}{25} \int \frac{x^4}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{36}{625} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx+\frac{52}{625} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{16}{125} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{16 x \left (2+x^2\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{1}{75} x \sqrt{2+3 x^2+x^4}-\frac{3 x \sqrt{2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac{16 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{26 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{625 \sqrt{2+3 x^2+x^4}}+\frac{3 \int \frac{62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{4375}-\frac{6}{625} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{1}{75} \int \frac{2+6 x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{3}{125} \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{16 x \left (2+x^2\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{1}{75} x \sqrt{2+3 x^2+x^4}-\frac{3 x \sqrt{2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac{16 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{23 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{625 \sqrt{2+3 x^2+x^4}}-\frac{3 \int \frac{-175-125 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{109375}+\frac{39 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{4375}-\frac{2}{75} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{2}{25} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{\left (3 \sqrt{1+\frac{x^2}{2}} \sqrt{2+2 x^2}\right ) \int \frac{\sqrt{2+2 x^2}}{\sqrt{1+\frac{x^2}{2}} \left (7+5 x^2\right )} \, dx}{125 \sqrt{2+3 x^2+x^4}}\\ &=\frac{6 x \left (2+x^2\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{1}{75} x \sqrt{2+3 x^2+x^4}-\frac{3 x \sqrt{2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac{6 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{2+3 x^2+x^4}}+\frac{44 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{1875 \sqrt{2+3 x^2+x^4}}+\frac{3 \sqrt{2} \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}+\frac{3}{875} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{39 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{8750}+\frac{3}{625} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{39 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{3500}\\ &=\frac{9 x \left (2+x^2\right )}{175 \sqrt{2+3 x^2+x^4}}+\frac{1}{75} x \sqrt{2+3 x^2+x^4}-\frac{3 x \sqrt{2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac{9 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{175 \sqrt{2+3 x^2+x^4}}+\frac{81 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{8750 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{44 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{1875 \sqrt{2+3 x^2+x^4}}+\frac{3 \sqrt{2} \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}-\frac{\left (39 \sqrt{1+\frac{x^2}{2}} \sqrt{2+2 x^2}\right ) \int \frac{\sqrt{2+2 x^2}}{\sqrt{1+\frac{x^2}{2}} \left (7+5 x^2\right )} \, dx}{3500 \sqrt{2+3 x^2+x^4}}\\ &=\frac{9 x \left (2+x^2\right )}{175 \sqrt{2+3 x^2+x^4}}+\frac{1}{75} x \sqrt{2+3 x^2+x^4}-\frac{3 x \sqrt{2+3 x^2+x^4}}{175 \left (7+5 x^2\right )}-\frac{9 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{175 \sqrt{2+3 x^2+x^4}}+\frac{81 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{8750 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{44 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{1875 \sqrt{2+3 x^2+x^4}}-\frac{39 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{12250 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}+\frac{3 \sqrt{2} \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.29148, size = 213, normalized size = 0.96 \[ \frac{-182 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+1225 x^7+5075 x^5+6650 x^3-945 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+135 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+189 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+2800 x}{18375 \left (5 x^2+7\right ) \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 177, normalized size = 0.8 \begin{align*} -{\frac{3\,x}{875\,{x}^{2}+1225}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{x}{75}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{13\,i}{2625}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{9\,i}{350}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{9\,i}{6125}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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